Editors: M. Thoma and W. Wyner
R.E Curtain (Editor) A. Bensoussan, J.L. Lions (Honorary
Eds.)
Analysis and Optimization of Systems: State and Frequency Domain
Approaches for Infinite- Dimensional Systems Proceedings of the
10th Intemational Conference Sophia-Antipolis, France, June 9-12,
1992
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
HongKong Barcelona Budapest
Advisory Board
L.D. Davisson • A.GJ. MacFarlane. H. Kwakemaak J.L. Massey-Ya Z.
Tsypkin "A.J. Viterbi
Editor
Honorary Editors
A. Bensoussan INRIA - Universit~ Paris IX Dauphine
L Lions Collage de France - CNES, Paris
INRIA Institut National de Recherche en Informatique et en
Automatique Domaine de Voluceau, Rocquencourt, B.P. 105 78153 Le
Chesnay, France
ISBN 3-540-56155-2 Springer-Verlag Berlin Heidelberg NewYork ISBN
0-387-56155-2 Springer-Verlag NewYork Berlin Heidelberg
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F O R E W O R D
The 10th Conference on "Analysis and Optimisation of Systems"
organised by INRIA is marked by a change in form : instead of
covering a wide range of topics, as in the past, this conference is
devoted to a specific domain of the theory of infinite- dimensional
systems.We plan to proceed in the same way in the future, and we
will cover the whole field of Analysis and Optimisation of Systems
by a series of specialized conferences.
Among the advantages, this allows the possibility of covering a
specific area in more depth and it gives the participants more
opportunity for fruitful interaction.
We would like to express our thanks to the Organizations which have
sponsored this Conference: DRET and ONREUR
We also would like to extend our gratitude to :
- the authors who have shown an active participation in this
Conference - the many reviewers who have accepted the difficult
task of selecting
papers - the chairpersons for having run with efficiency all the
sessions
of the Conference - all the members of the Organization Committee,
and particularly
Ruth Curtain who did a wonderful job. - The Public Relations
Departement of INRIA and particulary C. Genest
and F. Tapissier for their excellent work in making this Conference
really happen.
- Professor M. THOMA and the Editor SPRINGER VERLAG who have
accepted to publish this series in the Lecture Notes in Control and
Information Sciences.
A. Bensoussan J.L. Lions
This conference is under the sponsorship of:
DRET Direction des Recherches Etudes et Techniques ONREUR Office of
US Naval Research in Europe
INTERNATIONAL PROGRAMME COMMI'I"rEE
Chairperson R. CURTAIN University of Groningen /
INRIA-Rocquencourt
J. BALL J. BARAS L. BARATCHART J. BLUM F. CALLIER G. Da PRATO M.
DELFOUR P. GAHINET W. KRABS A.J. PRITCHARD M. SORINE J.P.
YVON
J. ZABCZYK
LOCAL ORGANIZING COMI'I"rEE
L. BARATCHART J. BLUM P. GAHINET M. SORINE J.P. YVON
INRIA-Sophia, France Universitd de Grenoble, France
INRIA-Roequencourt, France INRIA-Roequencourt, France
INRIA-Rocquencourt / Universitd de Technologic de Compi~gne,
France
CONFERENCE SECRETARIAT
INTRODUCTION
The aim of the conference was to bring together engineers and
mathematicians working in the field of infinite-dimensional systems
who are specialists in one or more of the following categories of
mathematical approaches :
_ semi-group approaches _ partial differencial approaches _
frequency domain approaches _ synthesis of state and frequency
domain approaches
Since these approaches use very different sophisticated
mathematical techniques, it is unusual for a scientist to be expert
in all of them. On the other hand, these different approaches all
purport to address the same control problems for the same classes
of linear infinite-dimensional systems. It is therefore important
to compare and discuss the advantages and disadvantages of these
different mathematical techniques. To help ameliorate the
communication gap a series of introductory tutorial lectures were
given by specialists in above-mentionned fields. The writter
accounts of these introductory lectures, together with their
references, form an extremely useful source of background
information on a very wide class of problems and approaches in
linear infinite-dimensional sys tems .
For example, the semi-group theme is represented by an article on
general background, an up-to*date survey on the Linear Quadratic
Control Problem and Riceati
equations and an account of the state-space approach to HOO-optimal
control problem. The results on this last very recent topic are
less maticre than those on the Linear Quadratic Control Problem,
but since the techniques involved are very similar a strong
interaction between these two domains of activity can be expected
in the future. Indeed,. it has already begun.
The second theme concerns input-output descriptions of systems in
terms of transfer functions.After an introductory article on
transfer functions, an outline of the coprime factorisation
approach to control synthesis for different classes of transfer
functions follows. The final paper on this theme concerns the
application of these techniques to the important problem of robust
controller design and in particular, finite-dimensional controller
design in these articles, the connections with the first theme
(semi-group approach) arc emphasized.
The third theme on partial differential equations is also
introduced with a background article on the Lions approach,
followed by recent results on Exact Controllability and
Stabilisation using both the Hilbert Uniqueress Method and high
frequency asymptotic methods.
The last theme on frequency domain approaches is also based on a
transfer function description, but the connections with a
state-space representation are not relevant.The first article
describes the utilization of classical ideas in Harmonic Analysis,
such as Hank¢l operators and Nehari's Theorem, to the systems
theory context. Robust controller design is treated in a more
general and deeper context in the second article and the
final
contribution to this theme surveys recent results on the same
HOO-domain viewpoint using techniques from Harmonic Analysis.
In addition, the proceedings contains several key survey papers on
fundamental topics such as geometric theory, robust stability radii
and relationships between input- output stability and exponential
stability. Last, but not least, the proceedings contains many
shorter contributions on recent original research covering all four
of the above- mentionncd themes.
VII I
Of course, bringing together scientist from four very different
approaches with the aim of having them interact with each other is
an ambitious one. In this we succeeded, especially during the final
round table discussion on "Different Approaches to Control of
Infinite-Dimensional Linear Systems". The aim of these proceedings
is to help to promote further interaction and fruiful collaboration
in the future between scientists working on different approaches to
the challenging problems in the field of infinite-dimensional
linear systems.
T A B L E O F C O N T E N T S
1. T U T O R I A L L E C T U R E S
S E M I - G R O U P A P P R O A C H
Introduct ion to Semigroup Theory A. J. PRITCHARD
...................................................................................................
Riccati Equations Arising from Boundary and Point Control Problems
I. LASIECKA
............................................................................................................
A State-Space Approach to the H**-Control Problems for I n f i n i
t e - D i m e n s i o n a l Sys t ems B. VAN KEULEN
.....................................................................................................
23
46
SYNTHESIS OF STATE AND F R E Q U E N C Y DOMAIN A P P R O A C H E
S
Infini te Dimensional System Transfer Funct ions F. M. CALLIER, J .
WINKIN
...............................................................................
72
Stabi l izat ion and Regulat ion of Inf in i te-Dimensional Systems
Using Copr ime Factor izat ions H. LOGEMANN
.........................................................................................................
102
Robust Control lers for Inf in i te -Dimensional Systems R. F.
CURTAIN
.........................................................................................................
140
P A R T I A L D I F F E R E N T I A L E Q U A T I O N A P P R O A C
H E S
Control for Hyperbol ic Equations G. LEBEAU
.................................................................................................................
160
An Introduction to the Hilbert Uniqueness Method A. BENSOUSSAN
........................................................................................................
184
F R E Q U E N C Y DOMAIN A P P R O A C H E S
The Nehari Problem and Optimal Hankel Norm Approximation N. J.
YOUNG
.............................................................................................................
199
Topological Approaches to Robutness T.T. G E O R G I O U , M. C.
SMITH
.............................................................................
222
Frequency Domain Methods for the H**-Optimization of Distr ibuted
Systems A. T A N N E N B A U M
....................................................................................................
242
2. CONTRIBUTED PAPERS
Disturbance Decoupling Problem for Infinite-Dimensional Systems H.
J. ZWART
..............................................................................................................
279
Simultaneous Triangular-Decoupling, Disturbance-Rejection and
Stabilization Problem for Infinite-Dimensional Systems N. OTSUKA,
H. INABA, K. TORAICHI
.............................................................
290
Robust Stability Radii for Distributed Parameter Systems : A Survey
S. TOWNLEY
...............................................................................................................
302
Solutions of the ARE in terms of the Hamiltonian for Riesz-spectral
Systems C. R. KUIPER, H. J. ZWART
...............................................................................
314
Regional Controllability of Distributed Systems A. EL JAI, A. J.
PRITCHARD
.............................................................................
326
On Filtering of the Hilbert Space-valued Stochastic Process over
Discrete-continuous Observations Y. V. ORLOV, M. V. BASIN
...................................................................................
336
Boundary Stabilization of Rotating Flexible Systems C. Z. XU, G.
SALLET
................................................................................................
347
SYNTHESIS OF STATE AND FREQUENCY DOMAIN APPROACHES
Frequency Domain Methods for Proving the Uniform Stability of
Vibrating Systems R. REBARBER
..........................................................................................................
366
The Well-Posedness of Acceleromctcr Control Systems K. A. MORRIS
..........................................................................................................
378
Comparison of Robustly Stabilizing Controllers for a Flexible Beam
Model with Additive, Multiplicative and Stable Factor Perturbations
J.B O N T S E M A , R . F . C U R T A I N , C . R . K U I P E R , H
. M . O S I N G A ............................ 388
On the Stability Uniformity of Infinite-Dimensional Systems H.
ZWART, Y. YAMAMOTO, Y. GOTOH
............................................................
401
×l
Modelling and Controllability of Plate-Beam Systems J. E. LAGNESE
..........................................................................................................
423
A Simple Viscoelastic Damper Model - Application to a Vibrating
String G. MONTSENY, J. AUDOUNET, B. MBODJE
.................................................... 436
Controllability of a Rotating Beam W. KRABS
..................................................................................................................
447
Min-Max Game Theory for a Class of Boundary Control Problems C.
MCMILLAN, R. TRIGGIANI
.........................................................................
459
Microlocal Methods in the Analysis of the Boundary Element Method
M. PEDERSEN
...........................................................................................................
467
Stochastic Control Approach to the Control of a Forward Parabolic
Equation, Reciprocal Process and Minimum Entropy A. BLAQUIERE, M.
SIGAL-PAUCHARD
......................................................... 476
Observability of Hyperbolic Systems with Interior Moving Sensors A.
KHAPALOV
..........................................................................................................
489
Controllability of a Multi-Dimensional System of Schr0dinger
Equations : Application to a System of Plate and Beam Equations J.
P. PUEL, E. Z U A Z U A . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
500
Decay of Solutions of the Wave Equation with Nonlinear Boundary
Feedback F. CONRAD, B. RAO
................................................................................................
512
Boundary Approximate Controllability for Semilinear Heat Equations
C. FABRE, J. P. PUEL, E. ZUAZUA
...................................................................
524
On the Stabilization of the Wave Equation O. MORGUL
................................................................................................................
531
FREQUENCY DOMAIN APPROACHES
The Hankel Singular Values of a Distributed Delay Line A Fredholm
Equation Approach L. PANDOLFI
............................................................................................................
543
Rational Approximation of the Transfer Function of a Viscoelastic
Rod K. B. HANNSGEN, R. L. WHEELER, O. J. STAFFANS
................................ 551
Some Extremal Problems linked with Identification from Partial
Frequency Data D. ALPAY, L. BARATCHART, J. LEBLOND
................................................... 563
XII
Approximat ion of Inf in i te-Dimensional Discrete Time Linear
Systems via Balanced Realizations and an Application to Fractional
Filters C. BONNET
..................................................................................................................
574
A "Relaxation" Approach for the Hankel Approximation of some Vibra
t ing St ructures N. M A I Z I . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
585
Numerical Methods for H** Control of Distributed Parameter Systems
D. S. FLAMM, K. K L I P E C
....................................................................................
598
Robust Controller Design for Uncertain Time Delay Systems Z. Q.
WANG, S. SKOGESTAD
.................................................................................
610
Parameter Ident i f icat ion of Large Spacecraf t Systems Based on
Frequency Character is t ics D. R. AUGESTEIN, J. S. BARAS, S. M. F
ISHER ........................................... 624
On the Optimal Minimax Tuning of Controllers for Distributed
Paramete r Sys tems S. P O H J O L A I N E N , M. LAAKSONEN
..................................................................
636
Introduction to Semigroup
Coventry CV4 7AL, UK
1 A b s t r a c t
In this paper some basic sytem theoretic concepts will be
introduced for abstract systems of the form
e(t) = Ax(t) + Bu(0, x(0) = ~0, y(t) = C.(t). (1)
Here A is the infinitesimal generator of a strongly continuous
semigroup S(t) on a Banach space Z and necessary and sufficient
conditions for this to be the case are given by the Hille-Yosida
theorem. For U another Banach space B E £(U, Z) and x ° • Z, u(.) •
L2(0, co; U) a mild solution is defined to be
j~0 t ~(t) = s(t)~ ° + s(t- s)Z~,(s)d~ (2)
and z(.) E C(0; co; Z). Various definitions of controllablity,
observability, stabi- lizability, detectability, identifiability
and realizability will be given and theorems which characterize
them will be stated. Throughout the paper examples will be given
(albeit trivial ones) which illustrate the way the abstract
definitions and re- sults can be applied to concrete problems
defined via pa~-tiaI differential equations and delay equations. In
preparing this introduction I have made considerable use of the
following book by Ruth Curtain and Hans Zwart An Introduction to
Infinite Dimensional Linear Systems Theory which is to be published
soon.
2 F o r m u l a t i o n o f t h e s y s t e m
Let z0 be the state, at time zero, of a linear, time-invariant,
uncontrolled dynamical system on a Banach space Z and z(t) denote
the state at Lime t. Then we may define a linear operator S(t) : Z
--* Z, such that
S(O) = I (identity on Z) (1) *(*) = S(*)~0. (2)
2
If the solution is unique, then z(t + a) is the same point in Z as
the point reached by allowing the dynarnics to evolve from z(a) for
.a time t, thus
z(t+~)=S(t+s)zo=S(t)z(s)=S(t)S(s)Zo, t>8>O, WoEZ
whence, the semigroup property
S(~+8)=S(OS(~), ~>_ ,>_0. (3)
If, for each t, the state z(t) varies continuously with variations
in the initial state, then S(t) is a bounded map. Finally, we
impose some smoothness with respect to time on the solution or
trajectory z(.). In fact, we only require z(t) --* zo as * --~ 0 +,
since this will imply that the trajectory is continuous for all
time. This motivates the following:
Defini t ion 2.1 (Strongly continuous semigroup) A strongly
continuous semigroup is a map S(t) from R + to £(Z) which
satisfies
s(o) = z (4)
s(~ + 8) = s(os( , ) vt,8 >_ o (~) I IS (O=o- =oll -~ o as ~ -~
o + , V=o e z . (6)
It follows immediately from this definition that there exists
constants M and w such that
I IS(~) l l - - M e ~' . (7)
However, since we have only assumed S(~)zo is continuous in ~, it
is not possible in general to differentiate S(f.)zo. Nevertheless,
we define:
Defini t ion 2.2 (Infinitesimal generator of a strongly continuous
semigroup) The infinitesimal generator A of a strongly continuous
semigroup S(t) on a Banach space Z is defined by
1
= lira ~ [ s ( o - z]~ (8) Az t-*O+
where the domain, D(A), of A is the set of z for which the above
limit exists.
Then, it can be shown that A is a closed, densely defined, linear
operator, and
d(S(Ozo ) = AS(Ozo = S(OAzo, zo e D(A) . (9)
Thus, z(~) = S(t)Zo is the solution of the abstract differential
equation
~. = A z , z(9) = Zo E D(A) . (10)
The Hille-Yosida Theorem gives a complete characterization of those
operators which generate strongly continuous semigroups.
T h e o r e m 2.3 (Hille-Yosida) A necessary and sufficient
condition for a closed linear operator A, with dense do- main in a
Banach space Z, to generate a strongly continuous semigroup S(t) is
~hat there exist real numbers M and w such that for all real A >
w, A is in the resolvent set of A, and
M II(,~i - A ) - ' I I < (A _ ~) - - - - - - - ; , r = 1 , 2 , .
. . (11)
i,~ ~,hich c~e IIS(t)ll <__ Me".
E x a m p l e 2.4 Let A E £(Z) , then
S(O = e ~' = ~ A"~"/n! n=0
is a strongly continuous semigroup on Z with infinitesimal
generator A.
(12)
E x a m p l e 2.5 Let {~o,},>1 be an orthonormal basis in a
sepazable Hilbert space H and (,~.)n_>l be a sequence of real
numbers. If sup A. < eo then
n
n=l
{h e n : Z ~=.(h, ~.>~ < ~}. n = l
We are part icularly interested in control problems which can be
wri t ten formally as
~(t) = Az(t) + Bu( t ) , z(O) = Zo (13)
where we assume B 6 £.(U, Z), U is a Banach space, and u 6 L2(0, T;
U). The first p roblem encountered is the concept of a solution of
(13). If Z is finite- dimensional, then (13) has the solution
f0 L ~(~) = s(t):o + s ( t - s)B~(s)ds. (14)
However, if Z is infinite-dimensional, we cannot in general
differentiate (14) to obtain (13). If this were the case, we would
say that (14) is a str /ct solution. Nevertheless, z(.) is well
defined by (14) and, in fact, z E C(0, T; Z). We call (14) the mild
solution of (13). And although our problems are mot ivated by
differential equations of the form (13), we will actually define
our system via the integral equation (14).
4
Finally, we will assume that the output from the system is related
to the state by means of the equation
y(~) = CzCt) (15)
where we assume C 6 £ ( Z , Y ) and Y is a Banach space. Then, from
(14),
f y(t) = cs(t) o + cs( - (16)
If zo = 0, note that
( s I - A)-~z = e - a S ( t ) z d t for Re s > w (17)
then, by taking the Laplace transform of (16), we obtain
9(s) = C(sI - A ) - ' Bfi(s) . (18)
Thus, the transfer function of the system is given by
G(s) = C(sl - A ) - I B . (19)
For many physical systems it is possible to obtain a different
representation which is important in the application of numerical
methods and approximation theory. In order to see how this can be
carried out, we specialize and assume that the operator A is
defined on a Hilbert space H, and has a distinct set of eigenvalues
{Ai}, with corresponding eigenvectors ~i such that {~Pl} span H.
Then,
A%ai = Ai~i. (20)
If lbi axe eigenvectors of the adjoint operator A', then
A'~bi = Ai¢i (21)
where Ai is the complex conjugate of Ai. Moreover,
(~bi ,~j)=0 if i # j (22)
and we may normalize ¢i, qal, so that
(~/,i,~ol) = i V{. (23)
Now, let us assume that there exists a solution of the form
z( t ) = (24) i = l
Substituting in (13), we find ~ = x d,(t)~, = Z~=I Aia,(t)~pi +
Bu(t ) . Taking the inner product with ~b, gives
a,Ct) = A,a, Ct) + (¢, , Bu( t ) ) , ,, = 1 ,2 , . . . (25)
We also have
oo
ai(O) = aio. (27)
sCt)z = f i e~'%(¢. ~>. (28) i = l
So we have seen that there are a number of differen~ ways of
describing our system. The differential equation may be given in
the form (13) or, alternatively, its mild so- lution may be
formulated in terms of the semigroup. A transfer function
description may be given or we may describe the system in terms of
an infinite set of ordinary differential equations. In order to see
how these different approaches are realized in practice, we
consider the following simple example.
E x a m p l e 2.6 Consider a metal bar in a furnace. The bar is
heated by jets whose spatial distribution b(z) is fixed but whose
magnitude (intensity) can be varied by a control u. For
one-dimensional flow, heat balance considerations over a small
region about the point z yield
= ~ ~ . - + b (z ) . ( t )d2
where 8(x, t) is the temperature in the bar at time t and distance
x from one end, a is the thermometric conductivity. Differentiation
yields
08 0~8" t) + b(x)u(t) (29) 0-7 = a0~z2 ~2'
Let us assume that the temperature at each end is kept at a
constant value T, and that the initial temperature distribution in
the bar is 8o, so that 8(0,t) = 8(l , t) = and 8(2,0) = 8o(x),
where I is the length of the bar. Set 8(2, t) = T ( z , t ) + "~
and 0o(2) = T o ( 2 ) + ¢ , then
OT O~T" t) + b(2)~(t) (30)
T ( O , t ) = r ( l , t ) = O (31) T(z,O) = To(z). (32)
Finally, assume that it is possible to sense the temperature at a
given point, and that this is accomplished by taking some weighted
average over nearby points. Then the output of
I the sensor can be modelled by h(t) = fo c(z)8(x, t)dx. Writing
h(t) = y(t) -F T f t o c(z)dx, then
f y(t) = ~ ( ~ ) T ( ~ , t ) e ~ . (33)
Equations (3O)-(33) represent a single-input, single-output
distributed parameter system. Let us define an operator A by
d2T (AT) ( z ) = a-~Tx2 (z ) (34)
D(A) = H=(O,l) n HI(O,O (35)
then we may abstract the differential equations (30/-(33) to
obtain
9.(t) = AzCt)+ buCt), z(O) = zo (36 / ~(~) = (~ ,K~ ) ) (37)
where z(t) = T( . , t ) , and the inner product is taken on
L2(O,l). It is easy to show that the operator A generates a
strongly continuous semigroup S(t), where
s(~)z = ~] ~-,--~-'~.(~.. ~) (38) n = l
with ~ . ( z / = (V/~7Osin nTrz/l, so that we can obtain the mild
solution (14/. In order to construct the transfer function, we have
to find (sI - A) -1, i.e. we have to solve the problem (sI - A)z =
~. for z E D(A) and any Y. E La(O, l). It is readily shown that the
solution is
1 [s inh(~ / ,~ ) ' /~=
['sinhC~l~l,/~Cl_ol~(~ld~_/o'sinhC.l,~ll/~C=_,.l~C,.ld,.] Z(Z) -
(,,,)'/= ' ~ .,o
so that the transfer function is given by
G(~) = (c, (~I - A)-Ib)
f0' c(~) [sinh(d~)~/2x [ sinh(8/c~)l/2(l a)b(~)da (~)w~ t
sinh(d~)ml J0
- /o" s~,~(~/,,)'/'(= - ,,)b(,,le,,]e=. (391
equivalent infinite-dimensional system, we assume that b(z) =
(40)
(41)
To obtain the oo ~"~'n=l b , . ~ . ( ) , c(x) = ~ = 1 c.¢2.(z), and
(z(t))(x) = ~ = 1 a.(t)~on(x), then
a . ( t ) = - l---~-a.(t) + b . ~ ( t ) , ~ = 1 , 2 , . . .
yCt) = ~ c . a . ( t ) .
3 Linear feedback control
In many appl icat ions , we wish to consider s ta te or output
feedback control, in which c a s e
~(~) = F z ( ~ ) (1)
o r
~(~) = ~yCt) = ~c~(t) (2) where we assume F E S(Z, U), ~' E £(Y,
U), Y is a Banach space. Note that (2) is just a special case of
(I) for which we have the following result:
Propos i t i on 3.1 Let A generate a strongly continuous semigroup
S(t) on a Banach space Z, B E E( U, Z) and F E ~.( Z, U), then A +
B F generates a strongly continuous semigroup S F( t ) which
satisfies
Z' s~ ( t ) z = s ( t ) z + s ( t - s ) B F S ~ ( s ) z e s
and g Ils(t)ll < M e ~'` then
IIS~(t)ll < M e (~+Mlla~H)' • (3)
A direct consequence of the above proposition is that z(t) =
SF(t)zo is the mild solution of the differential equation ~(t) = (A
+ BF)z( t ) , z(O) = zo. In some cases, we wish to consider
time-varying feedback operators of the form
,,(t) = F(t)z(~) . (4)
Obviously, in such cases the operator A + BF( t ) will not generate
a strongly con- tinuous semigroup; consequently, we introduce the
following generalization:
Defini t ion 3.2 (Mild evolution operator) Let A(T) = {(s,~) : 0
< s < t < T} then U(. , . ) : A(T) ~ / : ( Z ) is a mild
evolution operator if:
(a) U(t, t) = I Vt e [O,T], (b) U ( t , r ) U ( r , s ) = U( t ,~)
O < s < r < t < T , (c) U(., s) is strongly continuous
on Is, T], (d) U(t, . ) is strongly continuous on [0, ~].
A consequence of the above proprty is that
ess sup IIUCt, s)ll < ~ . C 5) a(T)
We have not assumed any differentiability properties of the mild
evolution operator; however, the following concept is useful.
Defini t ion 3.3 (Weak evolution operator) Suppose that for all t
> 0 A(t) is a linear operator on Z, its domain D(A) is dense in
Z, independent of t and A(.)z is continuous for every z E D(A).
If
or(t, ~)z - z = u(~, e )A (p )~ep ~, ~ ~ ~ ( T )
then U(f., s) is said to be a weak evolution operator with the
generator .4(-).
Note that
O u ( ~ , s ) z , = - U ( t , s)A(s)z0 s e [0,t], Zo e D(A) .
Now suppose that F E C(0, oo; £(Z, U)) then we have the
following:
P ropos i t i on 3.4 Let A generate a strongly continuous semigroup
S(t) on a Banach space Z, then
u(~, ~)z = s ( t - ~ ) . + s ( t - e ) ~ F ( p ) c r ( e , s )~ep
(6)
defines uniquely a mild evolution operator U(t, s). Moreover, U(t,
s) is a weak evo- lution operator with weak generator A + B
F.
4 Controllability
Note that if we require a strict solution of (2.13), then
necessarily z(t) E D(A) , t > 0. So if A is unbounded (i.e. D(A)
~ Z) it will be impossible to steer a strict solution to every
point in Z. For mild solutions we have the following:
Defini t ion 4.1 (Exact controllability on [0, T]) System (2.14) is
said to be exactly controllable on [0, T] if, given any two points
zo, zl E Z, there exists a control u E L2(0,T; U) such that
zl = S(T)zo + S (T - (1)
It can be shown that, if Z and U are reflexive Banach spaces, then
(2.14) is exactly controllable on [0, T] if and only if there
exists 7 > 0 such that
> 71{='llz.. (2)
E x a m p l e 4.2 We consider the controlled conduction problem of
Example 2.6 and ask whether or not it is possible to steer the
system, via controls u E L2(O,T), from the zero temperature
distribution to a prescribed temperature distribution T1 E L2(0, I)
in time T. Let TI(X ) = ~ . ° ° = l YnWn(X), where ~o,(x) = x /
(2~s in nrz / l . Then {vn} e l 2 and solving (2.40), we
require
Z r n = e;~"(T-')b,u(s)ds, n = 1,2,. . .
where .k, = -an2~r2/l 2. Using the Schwaxtz inequality,
I r . I 2 _< Ib . l 2 e2~"(T-°)ds u2(8)ds, n = 1,2, . . .
But, if the control energy is to be finite, then
I([b, I ~'lb, I 1/2
for some constants K and 2~'. So we see that the system cannot be
exactly controllable and only those states with "sufficiently
smooth" Fourier coefficients axe likely to be achievable.
o
The above example indicates that the concept of exact
controllability may be too strong for infinite dimensional systems.
In fact, it can be shown that systems gov- erned by parabolic
partial differential equations are never exactly controllable with
control action which occurs in practice (although this is not the
case for hyperbolic systems). Frequently, however, it may be
possible to control the system exactly to a linear subspace of a
Bane.ch space Z.
Defini t ion 4.3 (Exact W-subspace controllability on [0, T])
System (2.14) is said to be exactly W-subspace controllable if,
given any two points z0, zl, with z0 and zl E W C Z, there exists a
control u E L2(0, T; U) such that
/o T zt = S(T)zo + S ( T - s )Bu(s )ds .
If W, Z and U are reflexive Banach spaces, W C Z with continuous,
dense injection, S ( T ) W C W then (2.14) is exactly W-subspace
controllable on [0,T] if and only if there exists 7 > 0 such
that
IlB*S'(.)z*llL~to,r;v.) >__ 71lz'llw.. (3)
E x a m p l e 4.4 Consider the hyperbolic system
~ 2 " W . . at 2 (z, t) = (A)(x, t) + u(z, t) (z, t) e 12 x (0,
T)
Ow "x t" w ( x , O ) = - ~ ( , ) = 0 x e 1 2 (4)
~(~, t) = 0 (~,t) ~ 0n x (0, T)
where D C R a is an open bounded set. It is assumed that (¢nj) is
an orthonormal basis (in L2(D)) of eigenfunctions of the Laplacian
A with Dirichlet boundary conditions. The associated eigenvalues
A,~ have multiplicities r~. We may write (4) in the state space
form (2.13) [°l on the Banach space Z = Hol(a) x L2(a), with
Then A generates a strongly continuous semigroup S(t) where
r n
~ [/~,, ,.;> cos ~ -~om~ + ~_~o~1,~,,,,,~> ~o ~_~o~,~]
,o~
Z2
~__j~_.~[__(__)~n)l/2(Zl,¢nj)Sin((__~n)l]2~.}_~(Z2,¢nj)gos{(--~n)l/2~.}](/)nj
n j = l
We assume u e L2(0,T; L2(~)). Now
IlD*S'(t) [ zl ] 2 z~ ILL,( . ) =
Hence
+(z2, ¢-i) cos {(-;~.)z/2t}] 2.
i ~ ~ { _ ~o<z,.~o~>, [~ - sin ~ ~.~1,,T~1 . j=l 2(-'X")x/2
J
+2(_~.),/~(~, ¢.j)(~, ¢.j)[1 - cos (2(-~.)~/~T}]
+(z2, ¢.j)2 [T + sin ~{~(-~")l/2T} ]j}.
r n
11~l12z = ~ ~ [ - A.(~, 4,.j) 2 + (~2, ¢.s)2]. n j=l
10
So the system is exactly controllable if there exists 7 such
that
1 ~ "~ { sin {2(-A.)II2T} ]2(_A.) t12 J IT +sin {2(-~A.)112T} ]j -
~.<z,, ¢.j)~ [T - + <z2, ¢.i) 2
c~ {2(-;,.)InT} +2( - A")m(z~' ¢"#)(z:z' '/""J) 1 - 2(-A.)~n
l
r n
. j=l
This will certainly be the case if there exists 7 > 0 such
that
I. r r _ sin { 2 ( - ~ , , ) t / ~ r } _ 2 ~ 1 j + 2 X Y 1 - cos {2
( - .X , , )~ /~r} X 2 2(-A.) I I 2 2(--A.)I/2
+Y~[r + ~in {2(-~.)~I~r} _ ~;~] > 0 2(- ,x . )m
for X, Y E R. Such a 7 will exist if
T > sin {2(-An)l/2T}2(_An)II 2 II
and T2 [sin {2(-A.)II~T}] 2 > [i - cos {2(-A.)II2T}] 2
- - 4 . x . - ~ ( ~ I •
T > [sin {(-,X.) ' /~T} ( - ,~ . )m
which is valid for any T > 0. This result shows that for all T
> 0 the system is exactly controllable, m
Since stability is very often an important consideration, the
origin plays a distin- guished role.
Def ini t ion 4.5 (Exact null controllability on [0, T]) System
(2.14) is said to be exactly null controllable on [0,T] if, given
z0 6 Z, there exists u 6 L2(0, T; U) such that
- S(T)zo = S(T - s)Bu(s)ds. (5)
If U and Z are reflexive Banach spaces, then (2.14) is exactly null
controllable on [0, T] if and only if there exists 7 > 0 such
that
IIB'S*(.)z'IIL2(o,T;U. ) >_ 7}IS*(T)z*]Iz.. (6)
A weaker concept of controllability is the following:
11
Definition 4.6 (Approximate controllability on [0, T]) System
(2.14) is said to bc approximately controllable on [0, T] if, given
Zo, zl E Z, and s > 0, there exists a control u e L2(0,T; U)
with IIz(T) - zlU <_ s.
Although the assumptions of reflexivity of the Baaach spaces U, Z
axe not essential here, we will assume that they hold, then (2.14)
is approximately controllable on [0, T] if and only if
B'S*(~)z* = 0 V~ e [0,T] =~ z" = 0 . (7)
E x a m p l e 4.7 For Example 4.2, it is easy to show that S(t) =
S*($), where
S*(t)z* = f i e'X"t~2n(~p,,z*) and B'z = (b,z) n = l
and where An, £a,~ are defined in Example 4.2. Hence
B's'c)~* = ~ b,,e~"(~., ~').
Obviously, if b, = 0 for some n, then we cannot make any conclusion
about (£a., z ' ) and certainly cannot conclude that B*S*(t)z* = 0
implies z* = 0. So we require b, ~ 0. In fact, it can be shown that
this is a neccssary and sufficient condition for approximate
controllability, r~
D e f i n i t i o n 4.8 (Approximate null controllability on [0,
T]) System (2.14) is said to be approximately null controllable on
[0,T] if, given any z0 E Z and e > 0, there exists a control u E
L2(0, T; U) such that Ilz(T)]] < e.
It can be shown that a necessary and sufficient condition for
approximate null con- trollability on [0,T] is that the sct of
points for which
B'S ' ( t ) z* = 0 Vt e [0,T] (8)
is contained in the set of points for which
S ' (T ) z" = O. (9)
5 Observability
The input-output map of the system is given by (2.16),
namely,
f y(~) = cs ( t )~o + CS( t - . ) B ~ ( s ) d ~ .
If we assume that the control is known on the interval [0, T], then
so is the function
f ~(~) = y(f,) - CS(~ - s)Bu(s)ds (1)
12
and we have ~(0 = csCt)zo. (~)
Questions of observability are concerned with the problem of
determining the state of the system z(.) given the output y(.) or,
equivalently, .~(.). Clearly, we are able to compute z(t) for all t
>_ t if z(t) is known. Here, we consider two possible values of
t, namely 0 and T. Although it may seem from (2) that # E C([0, T];
Y), since we want to consider the possibility of perturbations to
the output we will take ~ E L2(0, T; Y) with Y a reflexive Banach
space.
Definit ion 5.1 (Initial observability on [0, T]) System (2) is
said to be initially observable on [0, T] if
c s ( O z o = o for a.~. t e [0,T] ~ zo = 0 . (3)
We see that if the system is initially observable then there is a
one-to-one relationship between the output and the initial state,
i.e. given g(.) and z0 satisfies (2) then z0 is unique. In many
physical problems this concept is not strong enough since, if the
output y is slightly perturbed, then the corresponding initial
state could vary considerably. We therefore introduce a continuity
hypothesis.
Definit ion 5.2 (Continuous initial observability on [0, T]) System
(2) is said to be continuously initially observable on [0,T] if
there is a continuous map between the output and the initial state,
i.e. if there exists 7 > 0 such that
I I~(Olla,(oz;~) = I lCS( ' )~ol l~=(oz~Y)>-~l lzo l lz-
(4)
We note immediately that the conditions (3) aad (4) are similar to
the control- lability conditions (4.7) and (4.2), respectively. In
fact, the concepts are dual to each other in the following sense.
Suppose Z is reflexive. Let us define our orig- inal system by the
sextuplet {Z, U, Y, S(t), B, C} and denote the dual system by {Z*,
Y', U*, S'(t), C*, B'}. Then, the original system is approximately
controllable on [0,T] if and only if the dual system is initially
observable on [0,T], and the original system is exactly
controllable on [0, T] if and only if the dual system is con-
tinuously initially observable on [0, T]. But we have seen that
exact controllability may be too strong a requirement, so the same
must be said of continuous initial observability. We have two
possible ways of preserving the continuity of the map fi'om output
to initial state. One way is to assume that the output is smoother
thaal that which we have previously considered, the other is to
seek a space larger than Z for which the continuity holds.
Example 5.3 Suppose Z = L2(0, co), to > 0
(sco~) (= ) o O < z < t
{ z ( z ) = __. to (C~) (z ) = 0 0 < = < to
13
Then
= LT fm~,~(to_t,o)lZ(P)12dP dr.
If T < to and z(x) = O, x >_ l0 - T, then f [ HCS(t)zH2dt = 0
and so the system cannot be continuously initially observable.
However if T > to, then
: Z '°/,o=_ > (m- to) --L ~ Iz(p)2dp
= (T - to)llz(-)ll~.,(o.~). Hence the system is continuously
initially observable.
Defini t ion 5.4 (Continuous initial V-large space observability on
[0, T]) System (2) is said to be continuously V-large space
observable on [0, T] if there exists a Banach space V such that Z C
V and the output to initial state map is continuous between L2(0,
T; Y) and V, i.e.
IlCS(.)~ollL=(oz;~,) > ;II;o11,,, (s)
for some 7 > 0.
Clearly, this is the dual concept to exact V*-subspace
controllability. Very often we require knowledge of the state z(.)
in order to construct controls. If we observe over an interval [0,
T], then the construction of controls may require z(t) for ~ >_
T, this leads to the following definitions:
Defini t ion 5.5 (Final observability on [0, T]) System (2) is said
to be finally observable on [0, T] if
CS(t)zo = 0 for a.e. ~ 6 [0,T] (6)
implies S(T)zo = O. (7)
Defini t ion 5.6 (Continuous final observability on [0, T]) System
(2) is said to be continuously finally observable on [0, T] if
there exists 7 > 0 such that
HCS(')ZollL~(o,r;r) >_ "flIS(T)zollz. (S)
14
6 Control labi l i ty and observabil ity in Hi lbert space
In many examples, U, Y, Z will be Hllbert spaces. Then if the
system is exactly controllable to the subspace W on [0, T], we
require
B * * 2 II s (')~lbco,r,~) -> ~11~11~. (1) Let us further assume
that there exists a 2 such that
a~ll~ll~. > lib S (')zllL, co,r;u) (2)
then
If we define the operator G by
f Gu = S(T - s)Bu(s)ds (4)
then G e £(L2(O,T; U), W), and
(G" z)Ct) = B*S*(T - t)z (5)
with G" 6 £(W*, L2(O, T; U)). Moreover,
/0 f C a " z = S ( T - s ) Z ~ B ' S ' ( T - s )~d~ = S ( t ) B B '
S ' ( t ) z a
and from (3), GG* • ECW', IV) and (GG')-' • £(W, W'). ~Ve define
the operator GG* to be the controllability operator W(T),
i.e.
W(T) = GG*. (6)
Now we axe in a position to construct a control which steers z0 • Z
to zl • W. Specifically, we require
zl = S(T)zo + S(T - s)Bu(s)ds (7)
with zl - S(T)zo • W by hypothesis. Let
u(t) = B*S'(T - t)W-'(T)[z~ - S(T)zo] (8)
then ll,,llt.,coz,u) <- IlV'll~c~.,~.,(o,~w))llW-'ll,-cw.w.)llz~
- S(T)zo[iw.
Thus, the control is well defined and it is easy to show that (7)
is satisfied. Similar arguments lead to a dual result on continuous
initial V-laxge space observ- ability. If we construct an operator
O(T) defined by
f O(T)z = S*(t)C'CS(t)zdt (9)
15
] : s'(t)C'y(t)e =
whence
Note
fi £ ( V , V ' ) . So, if ~(t) = CS(t)Zo, then operating by S ' ( t
) C " and
j~0 T O - I ( T ) S*( t )C'9( t )d t = zo. (10)
~00 T lizol[v <_ llO-1(T)Hr.(v.,v)Jl
S*(t)C'dt[[~(L,(O,T;Y),V.)I{~I[L,(o,T;y).
We have therefore obtained explicit formulae (8) and (10) for the
control steering Zo to zl, and for the estimation of the initial
state. In the first case, we require exact subspace controllability
and zl E W; in the second case, we do not necessarily expect the
initial state to be a Z-valued function. This may seem
unsatisfactory but the alternative is to assume that the output
data is smooth. However, if we are only interested in obtMning z
(T) , then we require continuous final observability and we have to
adjust (10) to give
S ( T ) O - I ( T ) S ' ( t )C '~ ( t )d t = z ( T ) . (11)
Similarly, if we require exact null controllability, then the
control is
u(t) = - B * S ' ( T - t)~Y -1 ( T ) S ( T ) z o . (12)
The advantage of such considerations is that, for parabolic
systems, S ( T ) is a smoothing operator so it may be that z (T )
as in (11) lies in Z not V; similarly, S(T)zo may lie in W. On the
other hand, for hyperbolic systems, where the semi- group is not
smooting, it may be possible to have W = V = Z (see Example
4.4).
7 Stability, stabilizability and detectability We have seen that,
for any strongly continuous scmigroup S(t) , there exist constants
M and w such that [[S(t)l [ ~ M e ~'t, t >_ O. We say that the
semigroup is exponentially stable if w can be chosen to be
negative. In this case, the solution z(.) = S(')Zo of the
uncontrolled system
~(t) = Az ( t ) , z(O) = zo
is such that I[z(t)ll <_ Me~'tllZoll and so decays exponentially
to zero. An important criterion for exponentiM stability is the
following:
P ropos i t i on 7.1 The strongly continuous semigroup S i t ) on a
Banach space Z is ezponentiaUy stable i f and only i f there exists
7 > 0 such that
fo llS(t)zll2d _< (Izll', Z. Z E
16
If Z is a Hilbert space the above result can be reformulated as a
Liapunov equation.
P ropos i t i on 7.2 Suppose that A is the generator of a strongly
continuous semi- group on a Hilbert space H. Then S(t) is
exponentially stable if and only if there exists a positive
operator P = P* E L(H) such that
(Az, Pz)H + (Pz, Az)H =--IIzi]2H, z e D(A) .
For finite dimensional systems a matrix A generates an
exponentially stable semi- group e A t if and only in sup{Re A : A
E a(A)} < 0. This is not necessarily the case in infinite
dimensions since the growth rate w is not necessarily determined
via the spectrum.
Defini t ion 7.3 We say that the semigroup S(~) satisfies the
spectrum determined growth assumption if
sup{Re A: A • a(A)} = l im{ln IIS( )ll/t} = ~0. (1)
(1) holds for a large class of semigroups and when this is the case
for every w > w0 there exists M~ such that lIS(l)ll < M~e ~L,
t > O. If S(t) is not exponentially stable it may be possible to
stabilize it by state feedback.
Def ini t ion 7.4 (A, B) is said to be stabilizable if there exists
F • L(Z, U) such that the semigroup SF(I) generated by A + B F is
exponentially stable.
One might hope that if the pair (A, B) is approximately
controllable then (A, B) is stabilizable. The following exaraple
shows that this is not necessarily the case.
E x a m p l e 7.5 Let Z = 12, U = C
Az = (z,,~212,z313,...,z,l~,...) ZU = (hi u , b 2 u , . . , , b n u
, , . . )
where b, ~ O, ~ = 1 Inb,] 2 < co. Then B e Z(C, Z) and it is
easy to show that (A, B) is approximately controllable. For any F e
~(Z,C) there exists f e 12 such that Fz = {z, f). Consider the
solutions of
Then
or
x . = z . / n + b . ( z , / )
z . = n x . - n b . ( z , f ) .
Now (nb,) E 12, but there exists x E 12 such that (nx,) ¢. 12. Thus
(A + BF) is not boundedly invertible and this implies 0 E a(A +
BF). m
One criterion which links controllability and stabilizahility can
be obtained via linear quadratic optimal control.
Propos i t ion 7.6 Suppose (A, B) is exactly null controllable on a
Hilbert space H, then ( A, B) is stabilizable.
17
In order to develop a more satisfactory theory of stabilizability
and also to ask converse questions like "what conditions on the
pair (A, B) are implied by the as- sumption of stabilizability" we
will assume the following:
Def ini t ion 7.7 If there exists a rectifiable, simple curve r
enclosing an open set a+(A) of a(A) in its interior and a(A) \a+(A)
in its exterior we say that the operator A satisfies the spectrum
decomposition assumption.
P r o p o s i t i o n 7.8 Suppose (A, B) is stabiIizable and B is
of finite rank then there exists 6 > 0 such that
a,(A) = a(A) N {A E C; Re k > 6}
consists of only parts of the point spectrum of A. Furthermore, for
evry )t E an(A) and every u > 0 dim k e r ( A I - A) v <
oo.
We see therefore that under the conditions of the above proposition
that A satisfies the spectrum decomposition assumption. If we
set
Pz = ~ / (AI - A)- 'zdA
Corresponding to this decomposition we sct
Z_ := (I- P ) Z .
a [A+ 0 ] 0 ] [B+] 0 A_ ' 0 S_(t) , B = B_ "
T h e o r e m 7.9 I f A generates a strongly continuous semigroup
on a Banach space Z and B is of finite rank, then the following
assertions are equivalent
(i) (A, B) is stabilizable, (ii) A satisfies the spectrum
decomposition assumption and there exists a r such
that Z+ is finite dimensional, S_( t ) is exponentially stable and
( A+, B+) is control- lable.
As a consequence of this theorem we recover the usual finite
dimensional character- isation of stabilizability "A is stable on
the uncontrollable subspace".
Def ini t ion 7.10 (A, C) is said to be detectable on a Banach
space Z if there exists K E £(Y, Z) such that the strongly
continuous semigroup SK(t) generated by A + KG is exponentially
stable.
If Z is reflexive then (A, C) is detectable if and only if (A*, C*)
is stabilizable.
18
8 Identif iabil ity
In this section we will be concerned only with parameter
identifiability, although it is possible to combine the ideas
introduced here with those of observability to develop the basic
structure for parameter and state estimation. We suppose that the
operators A, B, C, or equivalently S(t), B, C depend on a parameter
a; we denote this dependence by S°(~), B(o), C(o) and A(4). For
simplicity, we assume that o 6 £t C R", and define an ezperiment as
a pair [zo, u], denoting the collection of experiments by E = [z0,
u : u E /d]. Heze, we mean to imply that the initial state z0 is
fixed and a variety of experiments are conducted by varying the
controls in a sct/2. Of course, we could easily extend the
arguments to include a variety of initial states. The relationship
between the output and input is given by
io t
y(t,a) = C(o)S~(t)Zo + C(a)S~(t - s)B(4)u(s)ds. (i)
D e f i n i t i o n 8.1 (Indistinguishabili ty on [0, T]) The pair
of parameters 4, (7 6 £t are said to bc indistinguishable if y(t,
a) = y(t, 8) for a.e. t 6 [0, T] and for all experiments in E. If
this is not the case, then the pair is said to be
distinguishable.
Defini t ion 8.2 (Identifiablity on [0, T]) The parameter set gt is
identifiable at 8 if (8, 0) is a distiguishable pair for all
8,06~,4#8.
Although this is a desirable requirement, the mathematical analysis
associated with this concept is extremely difficult because the map
V(t, ") : f~ ~ Y, t 6 [0,T], is a highly nonlinear one.
D e f n i t i o n 8.3 (Local identifiability on [0, T]) A parameter
set ~t is said to be locally identifiable at 8 if there exists e
> 0 such that 8 ,4 is distinguishable for all 4 with I[0 - a]]R,
< ~.
Let us assume that Zo = 0, then clearly the set ~ is not
identifiable at 8 if for a # 8
o' [c(°)sa(* - ~)B(o) - cco)so(, - ~)BCo)NC,)a~ = 0 (2)
for all u E Zd and almost all t 6 [0,T]. Moreover, if the class of
control experiments is sufficiently large, (2) will hold if and
only if
c(4)sa(t)s(o) = c ( e ) s q O s ( o ) (3)
for almost all t e [0, T]. In general, it is di•cult to check
condition (3). However, if we set
19
where F(. , a) 6 £(U, L2(0, T; Y)) we may appeal to the imphcit
function theorem to check that the set f~ is locally identifiable
on [0, T] at 8. This requires that the operators depend in some
smooth way on the parameter a and that the Frechet derivative of
F(. , a) at 8 = a is invertible. The most usual way of tackling the
problem of identifying a is to construct a cost functional
J(~) = IIv(t,,~) - v.(t)ll~.e (4)
where y~ is the measured outcome of a single experiment. Then
assuming the initial state z0 is known the parameter a is chosen to
minimmize J ( a ) . Since the problem is highly nonlinear,
mathematical analysis is usually confined to imposing conditions
which ensure that the minimum exists and the major effort is
concentrated on ob- taining efficient computational algorithms.
Clearly, if y(., a) depends continuously on a E f14 a compact
metric space, then the minimum will be achieved. In order to
compute the minimum it is usually necessary to calculate the
Frechet differential of J ( a ) so extra connditions are assumed
which guarantee this derivative exists. There are at least two
problems with the above approach. First of all the initial state z0
is often not known and secondly the purpose of the identification
is to obtain a good model for a variety of inputs and not just
those used in the identification experiment. Below we suggest
methods by which these difficulties may be overcome. Suppose the
initial state is unknown and for simplicity the input function is
zero, then
Y(~, a) = CCa)Sa(OZo = (CoZo)(0 •
Assuming that for all a 6 f~ the system is large V-space
observable, we choose
Then (4) becomes
Then
~0 't v ( 0 = w ( t - , ) = ( , ) d , .
~0 T 7(~) _< I I C ( ~ ) s " ( t ) B ( ~ ) -
,.,,(OIl~(v,~.)dt
Y(~) = -<yo, ao(a:,a,,)-la~,v°) + Ily, ll= (6)
and a is chosen to minimize (6). Suppose now that Zo = 0 and we
wish to choose a E F/ so that the model (1) is reasonable for a
large class of inputs. One way of doing this is to choose a 6 ~ so
that
7(~) = sup j(~); II~(')II~,(o,~,~)= i (7) is minimized. But note
that we do not know in advance yc(~) for all inputs with
Ilu(')llL,(o,r;v) = 1. However it may be that from one input
experiment it is possible to obtain the input-output map in the
form
20
Y(,~) = sup IlC(,~)(/, , ,z - A(a))-XB(a) - wCi,,,)ll,~c~,,Y) • (8)
W
The RHS of this expression may be taken as a performance index for
the choice of a. We see that a variety of performance idices can be
associated with the identification problem. Just which or which
combination is the most appropriate depends very much on the
application.
9 R e a l i s a t i o n t h e o r y
There are two aspects to building a model of a system. One can
assume by using physical laws that the structure of the system is
given and the problem is then re- duced to one of finding the
values of parameters in the partial differential equations. For
complex systems it may not be possible to use physical laws to
model all of the system and a black box approach must be used. Here
controlled experiments are carried out which result in a knowledge
of the input-output map of the form
yCt) = ~oCt) + ~Ct- s)~Cs)~s. (i)
The question then arrises as to whether it is possible to construct
a state space model of the form
k(t) = Az(t) + Bu( t ) , z(O) = Zo e Z (2)
u(O = cz(o
and in what sense such a model is unique. This is the subject of
realisation theory. Of course we must interpret (2) in the mild
form
i' u(~) = cs(Ozo + c s ( t - ~)B~(s)d~. (3)
So that we require cs(Ozo = wo(O, cscoB = ~ ( 0 . (4)
In this section we will restrict our attention to the Hilbert space
c u e and Zo = 0, so that given w(t) we need to derive methods of
obtaining C, B and S(t) and exarnine the uniqueness of the
realisation of these operators. Clearly a necessary condition on
w(0 is
~,(~ + ~) = c ( O / - / ( s ) , tl s _> o (5)
where G(t) E £(Z , Y) and H(s) 6 £(U, Z). Given such a
factorisation we construct a canonical factorisation.
Definition 9.1 (Canonical factorisation) The operators G, H are
said to be a canonical factorisation of w if
AkerC(0 = {0}, nkerH'(s)= {0}. (6) t>0 °>_0
21
w(t + s) = CS(t)S(s)B, G(t) = CS(t), H(s) = S(s)B (7)
then a canonical factorisation will yield a realisation which is
approximately con- trollable and initially observable on [0, oo).
To construct a canonical realisation we set
M = ~'] ker G(t) the unobservable space (8) t>o
N = ( ' ] ker H ' ( s ) the uncontrollable space (9) a_>0
and set Z' = M ± Cl N j" which is a Hilbert space with the topology
induced by Z. We then denote by Pz, the orthogonal projection of Z
onto Z' and by ~r = P~, the injection of Z' into Z. Then we define
G'(t) =- G(O~r , H'(s) -= Pz, H(s) and it is easy to show
that
w(t + s) = G'(t)H'(s), s , t > O
is a canonical factorisation. Since we have a canonical
realisation, the operator
Oz = e-"'G'*(p)G'(p)zdp (10)
is well defined for some w > 0 and is 1-1 and selfadjoint. If we
now assume
Z o s ( t ) ~ = o -~ e-~"c"(e)a'(p + t)~eo (11)
defines a strongly continuous semigroup on Z', then there exists a
realisation with
C = a ' (0) , B = H'(0). (12)
In order to study the relationships between two canonical
realisations of the same w(.) it is necessary to strengthen the
definition of canonical. In fact we will assume that a realisation
C, B, S(t) on Z is exactly controllable to the subspace W and
continuously observable to the larger space V, with the
corresponding assumptions on C, B, S(t) on 2 which is another
realisation of w(t). Then if
C S ( t ) B = ~(t)~ (13) ^ A
it follows that there exist operators M, N, M, N such that
C = CN B = M B S(t) = M S ( t ) N (14) ¢ = c]~ b = PZB $(t) = ~ s c
t ) ~
N E £(W,I~.), M E £(V, V), IV E #(IV, W), 1(¢ E £(V, (/), M N is
the injection W ~ V, M N is the injection W --* V and
S(t)]V -= MS(t ) $ ( t )N = ~IS(t) . (15)
22
The essential idea behind the above is illustrated as follows
o * e-~,( t+~)S,( t )C,CS(t)S(p)S(7)BB.S,(7)zdtd7
= foz*e- 'Kt+~)S'( t )C*dS(t)sCp)SCT)~B'S'(7)zdtd7
by (13). Thus
W~z = fo~"e-~"S(t)BB'S'(t)zdt.
Then M = 0 - 1 0 1 , N = YVIYV -1 and similar definitions for the
other terms. Note that the operators M, N etc. are not bounded from
Z to Z unless, of course, V = W = Z, V = 1~ = Z which is the case
for exact controllability to Z and continuous initial
observability. The converse result is also valid, namely if (14)
hold then the realisations (C, S(t), B), (C, S(t), B) give the same
input-output map. In most applications w(t) will not be known for t
> 0 and we have to extend w(.) on say [0, T] either by analytic
continuation or periodicity.
Riccati equations arising from boundary and point control
problems
Irena Lasiecka Department of Applied Mathematics
University of Virginia Charlottesville, VA 22903
Abstract
We present a survey of results on differential and algebraic
Riccati equations which include the cases
that arise from boundary/point control problems for partial
differential equations (P.D.E.'s). As the
Riccati theory rests on dynamical properties of the underlying
P.D.E.'s (such as regularity, exact
controllability, stabilization, etc.), particular emphasis will be
paid to these. To this end, P.D.E.
methods (including pseudo-differential techniques and mierolocal
analysis) will be emphasized. The
paper will highlight an interplay between semigroups or operator
methods and P.D.E. techniques. This
survey is an update of the recent Springer-Vedag volume
[L-T.13].
1. Introduction
1.1 Classical theory: B bounded.
Let Y, U, Z and W be Hilben spaces. Let A: Y ~ D (A) ---> Y be
the generator of a s.c. sernigroup eat
on Y: t > 0 and B: U -.> Y be a given linear bounded (for
now) operator. The classical linear quadratic
optimal control problem consists in finding u ° e I.~(0,T; U)) and
y0 e La(0, T; Y) such that
(1.I) J (u 0, y°(u°)) = rain J (u, y(u)) u ~ I.~(O.T; U)
where the quadratic functional J(u, y) is defined by
T 2+lu(t)12 dt+lGy(T)12 w (1.2) J ( u , y ) = ~ [IRy(t)l z u
]
o
and y (u) is the solution due to u of the dynamical system
(1.3) y t = A y + B u ; y ( 0 ) = y o ~ Y .
Here R (resp. G) are bounded linear operators from Y --> Z
(resp. Y ---> W). In (1.2) T > 0 may be finite
or infinite. I f T = ~o we shall then drop the last term with G in
(1.2).
It is well known that there exists a unique optimal solution (u 0,
y0) to the minimization problem (I.I),
under an additional finite cost/stabilizability condition if T =
~,. The problem of interest in control
theory is to find a pointwise feedback representation of the
optimal control. This amounts to finding the
operator say, C (t); Y ---> U, independent on Yo, and time
independent in the case T = ~, such that
(1.4) u°(t; Y0) = C (t) y°(t; Y0) a.e. in 0 < t ~ T .
24
The advantage of having a "closed loop" feedback control is well
known and documented, with
motivation coming from engineering problems. Indeed, the
fundamental reason for using a feedback
representation is to accomplish performance objectives in the
presence of uncertainty. In many
situations, knowledge of the system is only part~; or else, the
available model is based on many
simplifying assumptions which question its accuracy. An effective
feedback reduces the effects of
uncertainties, because it tends to compensate for all errors,
regardless of their origin. It is well known,
at least since the work of R. Kalman in the early sixties in the
finite dimensional case, that the existence
of the feedback operator C (t) is closely related to the
solvability of the following Riecati Equations, for
the two cases T < ,,o and T = ,~, respectively.
Case T < o=: the Differential Riccati Equation in the unknown
P(t) E £, (Y), for all x, y e D (A)
d (DRE) ('~'t P(t) x, y)y = (A*P(t) x, y)y + (P(t) Ax, y)y + (Rx,
Ry) z - (B* P(t) x, B*P(t)Y)u; P(T) = G*G.
Case T = oo: the Algebraic Riccati Equation in the unknown P e L
C/), for all x, y E ~ (A)
(ARE) (A*P x, y)y + (PA x, y)y + (Rx, Ry)y = (B* Px, B* PY)u
"
It is well known that for the classical linear quadratic control
problem (i.e. when all the operators B, R,
and G are bounded), a unique solution P(t) e L (Y) to the DRE
equation for T < oo [respect. a unique
solution P e £ (Y) to the ARE equation for T = ~,] exists in the
class of nonnegative selfadjoint
operators [under the appropriate finite cost (stabilizability) and
detectability conditions, if T = oo], in
which case the operator C (O in (1.4) has the specific form
(l.Sa) C( t )=-B*P(t ) , henceu°( t ;yo)=-B*P(Oy°( t ;yo) a.e. 0
< t < T
(l.5b) [C=-B*P, henceu°(t;yo)=-B*Py*(t;yo) a.e. 0:~t<oo].
Moreover, in the infinite horizon case (i.e. T = oo), the feedback
-B*P in (1.5b) constructed via the
Riccati operator yields a closed loop dynamics yt ° (t; 3'o) = (A -
BB*P) 3,0 (% Yo) which is exponentially
stable; i.e. there exist constants M, 03 > 0 such that
(1.6) le (A-BB'p)' Iz(y) <Me-t°t; t > 0 .
Thus, in the case T = 0% when the original dynamics e At is
unstable, the Riccad feedback in (1.Sb) has
the additional attractive property of inducing uniform stability of
the closed loop optimal dynamics. All
the results mentioned above are suitable extensions of finite
dimensional theories (going back to R.
Kalman), combined with basic theory of linear semigroups (see
[B.1]; [C-P]; [B-P]; [Lio.1]).
1.2 More recent theories
In more recent years considerable attention has been paid to
control problems with data, i.e. operators R,
G and B, which do not satisfy the usual regularity assumptions.
Here, the main motivation comes from
2 5
boundary/point control problem for partial differential equations
with boundary/point observations.
Indeed, in some systems, for physical and technical reasons, only
the boundary of the spatial domain, or
else some selected points in the interior, may be aecessible to
external manipulations by actuators and
for sensing. From a theoretical standpoint, boundary/point control
problems pose much more difficult
questions than those with "distributed" controls. The main
mathematical feature that distinguishes
boundary/point control models from distributed models is the fact
that in the first case the control
operator B: U ~ Y is unbounded. In fact, in this case, the control
operator B is defined only in a
"larger" space i.e.:
(1.7) B e L(U;[~D(A*)]'), equivalently ( 'Ao-A) -1 B e £,(U;'Y);
Xo~ p(A)
where [D (A*)]' is the dual (pivotal) to D (A*) with respect to
Y-inner product; p (A) is the resolvent
set. A consequence of (1.7) is that the operator mapping the con~ol
into the state space may be
unbounded. Moreover, and more importantly, the quadratic term in
the Riccati Equation involving the
gain operator B*P may become unbounded (see below). This, of
course, complicates substantially the
mathematical analysis of the problem where standard methods of
proving existence of solutions to
Riccati Equations arc no longer applicable[ The main goal of this
paper is to present a survey of recent
results, thus updating [L-T.13] on the solvability of Riccati
Equations which arise in "nouregnlar"
control problems i.e. when the operator B is unbounded and subject
only to (1.7), as in [L-T.4]. In an
analogous Way, we can also treat the nonregular problems where the
observations R and G arc
unbounded, or where both control and observations are unbounded.
For lack of space, since the results
with nonregular observations arc somewhat simpler than those with
nonregnlar controls (and can
certainly be obtained by the techniques dealing with nonregular
inputs), we shall concentrate only on
the latter. Thus, in the sequel, we shall consider explicitly only
the case where the control operator B is
unbounded as in (1.7), while the observation operators R, G axe
bounded. Then, in order to ¢xu'act "best
possible" results in boundary/point control problems, it is
n~essary to distinguish different (not
necessarily mutually exclusive) classes of dynamics. One class
includes free dynamics which generate
s.c. analytic semigroups ¢ At on Y: t > 0. Canonical examples
include not only heat/diffusion
equations, but also damped wave/beam/plate equations with a
sufficiently strong degree of damping.
This class will be analyzed in section 2 below (theory) and in
section 3 (examples). It will be seen,
among the wealth of results available, that the gain operator B*P
is always bounded on Y in this case, a
result which reflects and contains the property that in this
analytic case P is a smoothing operator.
Another class to be treated in section 4.1 (theory) and in section
5 (examples) is motivated by, and
includes, wave, plate, and Schrodinger equations. It is
characterized by the property (in addition to
(1.7)) that the input-solution operator is wall-defined in Y. This
property is equivalent, by duality or
transposition, to assumption (H.2) in section 4.1, which is an
"abstract trace property." It is precisely in
the latter dual form (H.2) that this property has been established,
over the past ten years, for a large
variety of partial differential equations by purely p.d.e methods
(differential and pseudodifferential).
For this second class (H.2) of section 4.1, very different
techniques arc needed in the study of Riecati
equations, as compared to those used for the analytic class (H-l)
in section 3. In particular, the operator
2t~
P is now typically an isomorphism on Y (unlike the analytic class
(H.1)), and moreover B*P is now then
inherently unbounded in the most representative dynamics of this
class (H.2), such as conservative
problems. We shall see in section 4.1 that there is a link between
the desirable property of the feedback
B* P to yield the exponential decay (1.6) of the (optimal) feedback
semigronp and the unboundedness of
B*P.
FinaLly, in subsection 4.2, with motivations coming again from wave
and plate boundary control
problems (illustrated in section 6), we shall consider a very
general class where the continuity of the
input-solution map is not fulfilled, (i.e. (H.2) is violated). In
this latter class, P has generally unbounded
inverse and moreover, in the full generality of the present
assumptions, new pathological features make
now their appearance, such as the optimal feedback dynamics, which
is now claimed to be only a one-
time integrated semigroup, rather than a bonafid¢ s.c. scmigroup.
F'mally, section 7 deals with more
general "nonstandard" Riccati equations where the quadratic term is
nonpositive and unbounded. These
types of equations arise in the context of game theory and in
particular of H" theory.
2. Analytic semigroups.
The main hypothesis assumed throughout this section is
(H-l) thes.c, semigroup e^t is analytic on Y, t > 0 a n d A ~ B
E f. (U; H) for some 0 < y <l
where A = (7,.0I - A); 7,0 e p (A).
Examples of dynamics complying with this hypothesis are heat
equation with Dirichlet or Neumann
boundary control, strongly damped plate equation with point or
boundary controls, etc. (see sect. 3).
It should be noted that the main technical difficulties in the
analytic case arise when the constant 3¢ in the
hypothesis (H-l) is greater than ~ . Indeed, if ~ < ~,,~, then
the solution operator u --* y(t) is bounded:
L2(0,T; U) ~ C ([(3, "17; Y). This fact together with the analytic
estimates of the semigroup yields a
priori bounds for the gain operator B*P. The situation is more
demanding if, instead, T > ½, in which
case the solution operator is not bounded (even for 1-dimensional
problems) in the above sense. In
order to obtain a meaningful definition and eventually a priori
bounds of the gain operator B*P (hence
of the quadratic term in the Riccati Equation), much more refined
arguments involving the theory of
singular integrals are used.
2.1. Differential Riccati Equation
The results and the regularity of the Riccati operator depend on
the degree of smoothness of the
"terminal" operator G ~ f~ (Y; W). We first deal with the case when
the "terminal cost" operator
satisfies the following smoothing assumption
(2.t) A"I'G*GE L 0 0 , 'g as in (I-I.1) .
27
Theorem 2.1. (see [L-T.I], [FAD
Assume (H-I) and (2.1). Then there exists a nonnegative selfadjoint
operator P(t) = P*(t) > 0 such that
(2.2) P(t) ¢ L (Y; C ([0, T]; Y)) ;
in fact, even more for any 0 < 0 < I
"*0 CT (2.3) I A P(t) l £ (Y) < (T - t) ° ---'---~ "
(2.4) B*P(-)¢ L(Y;C[0, T];U) and the synthesis in (1.4) holds for
aU t ¢ [0,T].
(2.5) For 0 < t < T; P(t) satisfies the (DRE) equation for
all x,y ¢ D (,~); V 8 > 0.
(2.6) lira P(t)x=G*Gx; x¢ Y. t.--~T
(2.7) The solution P(t) is unique within the class of positive
selfadjoim operators such that
(2.4) holds.
(2.8) (regularity of optimal solutions): u 0 ~ C ([0,T]; l.D; yO ¢
C ([0, T; U). Moreover u ° and y0 are
infinitely many times differentiable on (0,T).
Notice that in the case when the operator G is subject to the
"smoothing" hypothesis (2.1), the optimal
control, the optimal trajectory, and the feedback operator B*P(t)
are regular (in fact, continuous) for
0 < t < T. Instead, in the absence of the smoothing condition
(2.1), the gain operators C(t) -- - B 'P( t ) as
well as the optimal control and trajectories develop singularities
at the terminal point t -- T. This is not
surprising, since the benefit of analyticity of the original
dynamics can not be relied on at the end point t
-- T. The next theorem provides the results pertinent to the
general "nonsmoothing" observation G. We
shall let I11.: L2(0,T; U) ---r Y be the (unbounded) operator
T L T u = t eAfr- t) B u(t) dt
with densely defined domain ~9 (LT) = {u ~ L2(0,T; U); I.: r u ¢
Y}. Clearly, L T is closable.
Theorem 2.2. [L-T.1]:
(2.9) GLT: L2(0 ,T;U) ~ W isc losab le .
28
Then there exists a non negative selfadjoint operator P(t) = P* (t)
such that
(2.10) P (.) • : (Y; C ([0,T); Y)
and the (DRE) equation holds as in (2.5). Moreover
cT (2.11) l(,~')°P(t)l m <--~- ~ , o<e<l,
CT (2.12) IB*P(t)IL(y;u) < (T-t)'f ' 0<t<T,
(2.13)
(2.14)
where
0<t<T,
lira (P(t) x, y)y = (G*G x, y), V x, y • Y, z...cT
U ° • C.f ([0, "r]; U); y0 • C2r-i+, ([0, T]; Y),
CT ([0,T]; ~0 "" [f(t) e C ([0,T); Y); t f l C,(10.T]; ",9 = teS~
(T-t)r I f(0 t Y } .
Moreover, the optimal control and trajectory arc differcndablc
functions of t for t e (0, T). []
Remark 2.1. It is shown in [L-T.I] by examining further an example
in IF.3] that without assurapdon
(2.9). the optimal control does not exist.
Remark 2.2. A sufficient condition for (2.9) to hold is
(2.15) (,~*)It~ O °
bc densely defined as an operator W ~ Y for sorac ~ > 2 T- I.
Notice that if Y <: lh then condition (2.15)
is automatically satisfied. For other related results on this topic
see [L-T. I] and [F.2].
2.2. Algebraic Riccati Equations
In this subsection wc shall then discuss the solvability of the
Algebraic Riccati Equation (ARE). A
necessary and sufficient condition for the existence of the optimal
control corresponding to the
rainiraizadon praoblcra (I. I) with T -- oo is the following
(F.C.C.) Finite Cost Condition: For oven/ Y0 • Y there exists u •
L2(0,0o ; y) such that the
corresponding value of the functional in (1.2) satisfies J (u,y(u))
< oo.
29
I. Existence
Assume hypotheses (H-l) and (F.C.C.). Then there exists a
selfadjoint, nonnegative definite solution
P = P* • L (Y) of the (ARE) equation such that
(2.16) (~k*) t - t p • r .(Y), V e > 0 , inpardcular
(2.17) B*Pe L ( Y ; U )
(2.18) For each fixed Y0 • Y, we have y0 (t; Y0) = e(A-BB'P)t Y0,
where the s.c. semigroup e (A-sB" p)t
is analytic on Y.
II. Uniqueness. In addition to the assumption of part I, we assume
the following "detectability
condition."
(DC) There exists K • L (Z; Y) such that the S, C. semigroup e
(^+zR)t generated by A + KR is
exponentially stable.
(2.19) Then, the solution P to (ARE) is unique within the class of
non-negative selfadjoint operators in
L (Y) which satisfy (2.17).
(2.20) The s.c. analytic semigroup e Art generated by Ap = A - BB*P
is exponentially stable on Y. •
Remark 2.3. Notice that in the analytic case the Riccati operator P
has a "smoothing" effect (inherited
from the analyticity of eat). Indeed, (2.16) asserts that P is
bounded from Y into ~)(/~*l-r). If the
resolvent of A is compact, this implies compactness of P in L
(Y).
3. Examples illustrating the results ofsection 2
3.1. Heat equation with Dirichlet boundary control [L-T.13, p.
51].
Let ~ c R n be an open bounded domain with sufficiently smooth
boundary F. In ~ , we consider the
Dirichlet mixed problem for the heat equation in the unknown
y(t,x):
Yt = Ay + c2y in (0,T] × f~ -= Q ;
(3.1) Jy(0, • ) = yo in f~ ; I
[ylz =u in (0,T] x r-- Z;
with boundary control u e L2(~) and Yo • L2(~). The cost functional
which we wish to minimize is
then
T
i fT < oo ,o r in the case T = oo
30
(3.3) J(u'Y)= i [Y(t)12ta + lu(t)l~]dt
where we denote I y I n = l y I In(n) and I u I r -= I u I Lz(r)"
To put problem (3.1) into the abstract setting
of section 2, we introduce the operator
(3.4) A h = A h + c 2 h ; a ) ( A ) f H 2 ( ~ ) m H 1 ( f 2 )
'
next extend isomorphicalIy A as L2(f2) -+ [~D (A*)]' and select the
spaces
(3.5) Z = Y = W = L2(f2); U = I..2(1"3,
and finally define the operators Bu - - ADu; R = I; G = I where D
(Diriehlet map) is defined by
h = D g i f f ( A + e 2 ) h = 0 i n ~ a n d h l r f g .
It can be shown (see [L-T.13 p. 52]) that hypothesis (H-l ) is
satisfied with 7 = ¾ + e where e > 0.
Moreover, since the condition (2.15) holds trivially with G = I,
hypothesis (2.9) is satisfied as well.
Hence, the conclusions of Theorem 2.2 apply. In the ease of
infinite horizon problem (T = **), the
arguments of IT.l], [L-T.13 p. 52] provide a construction of
stabilizing feedbacks, which in turn,
implies the Finite Cost Condition (F.C.CT. The detectability
condition holds automatically true since R
-- I. Thus, all the hypotheses of Theorem 2.3 are satisfied and the
statements (2.16)-(2.207 are valid for
problem (3.17 - (3.3).
3.2. Structurally damped plate equation with point control [L-T.13,
p. 57].
Consider the following model of a plate equation in the deflection
w (t, x), where 13 > 0 is any constant
wtt + A2w - pAwt = 8(x - x 0) u(t); in (0,T] × f2 - Q
(3.6) ~w(O, • ) = wo; wt(O, • ) = wl in f2 /
[wly =,Xwly =o in Z
where x 0 is an (interior) point of ft , dim ~ = n. The cost
functional associated with (3.6) is
(consistently with IT.2])
T (3.7) J ( u , w ) = t [IAw(t) l ~ + l w t ( t ) l ~ + l u ( t ) l
~ . ] d t + l A w m l ~ -
To put problcm (3.6) (3.77 into the abstract setting, we introduce
the strictly positive definite operator
. ~ h = A 2 h ; ~ ( . ~ ) = { h E H 4 ( ~ ) ; h l r = 0 ; A h l r =
0 } ;
and select the spaces and operators
Y = D (~'~) x L2 (~ ) = [H2(f2) n n~(f~)] × L2(t2) ; W = Z = Y ; U
= R t ;
31
I -°, 'I ,, I -° I I°I • o I °l A = -pA ½ ' 8(x x0)u ' ' "
n It can be verified see (sea [L-T.13, section 6.3] that the
hypothesis (H-l) is sadsfiexi with "/= ~-+8,
which then requires n ~; 3. As in the previous case, one verifies
that the condition (2.15), hence (2.9),
holds true. Thus Theorem 2.2 applies. Similar analysis applies to
the infinite horizon case (see [L-T]
scct. 6.3) where now Theorem 2.3 applies.
3.3. Structurally damped plate equation with boundary control.
[L-T.13, p. 64]
Consider
(3.8) w(0 , ' )=wo;wt (0 , ' )=wl in £2;
wlz=0; AwIZ=ue L2(0, T; I.a(ID).
T with (3.8)we associa,e the functional ,(u.y>= I t,~w~,), ~ +
,u~0 ,~ ld ,+ ,w~ ,~ To put
problem 0.8) into the abstract setting, we introduce Y, W, Z, A as
in example 3.2: U = I.,2(1") and the
operator B I°l BU ~ - ~ Du
It can b~ shown (sea [L-T.13, p. 65]) that hypothesis (H-I) holds
with 1:ffi ¾ + ~ Since condition (2.1)
is satisfied, the conclusion of Theorem 2.1 applies as well.
4. Hyperbolic and "hyperbolic-like" dynamics
Here we shall consider "unbounded control" dynamics governed by
general C0-semigroups, with
particular emphasis on hyperbolic and "hyperbolic-like" dynamics
(waves and plates). Because of space
limitations, we shall focus only on infinite horizon problems and
corresponding Algebraic Riccati
Equations. The results on finite horizon problem (Differential
Riccati Equations) can be found in [D-L-
T], [L-T.2], [L-T.3], [L-T.13] and references therein. A main
reason why Algebraic Riccati Equations
are particularly interesting is that the existence of solutions to
these equations is closely related to the
problem of stabilizability of the original, usually unstable
dynamics. On the other hand, questions of
stabilizability/controllability for hyperbolic (hyperbolic-like)
equations present many challenging
mathematical problems and have atwacted in recent years a lot of
attention. We shall make an effort to
enlighten this interplay between Riccafi theory and smbilizability
pmpe~es of the underlying dynamics.
32
4.1 Unbounded control operators subject to (H-2) hypothesis In this
subsection we present results on
solvability of Algebraic Riccati Equations in the case when (1.7)
holds and the unbounded control B
operator satisfies the following hypothesis
T (H-2) [ tB* e A't y 12 d t<CT lyl 2" u y, ye ~D(A*); extended
to all y ~ Y
where B" e r.(~)(A*);U) and (B* v, U)u =(v, BU)y; ve ~D(A*); ue U.
As we shall see later,
hypothesis (H-2) in the case of hyperbolic p.d.e.'s, expresses
certain "wace regularity" properties of the
homogeneous problems (see section 5).
Theorem 4.1. [L-T.2] [F-L-T]
I. Existence Assume that, in addition to the (F.C.C) and (1.7), the
regularity hypothesis (H-2) holds. Then there
exists a nonnegative solution P = P* e L ('Y) of the (ARE) such
that
(4.1) B*Pe LCD(A); U);
(4.2) uO(t;yo)=-B*Py°(t;yo)e I~(0, oo;U);
y0(t;y0)=e(A-BB'P)ty0;
(4.3) j(u 0, y0) = (pyO, y0)y, YO e Y.
when the operator Ap mA-BB*P is closed, densely defined with
~9(Ap)cY and it generates a
semigroup on Y.
II. Uniqueness
In addition to the hypotheses of part I, we assume that the
detectability condition (D.C.) holds true with
an operator K satisfying
(D-CA) IK*xl Z < C[IB*xl U + {Xiy].
Then the solution to (ARE) is unique within the class of
selfadjoint, positive operator satisfying (4.1).
Finally, e Apt is exponentially stable on Y Ill
As mentioned in the introduction, it is important to notice that,
in contrast with the analytic case
described in section 2, the gain operator B*P is now generally
unbounded. Indeed, this property follows
from the next results.
Theorem 4.2 [F-L-T] In addition to the hypotheses of part I of
Theorem 4.1, we assume the following exact controllability
condition:
33
(E.C.) The equation Yt = A* y + R*v is exacdy controllable from the
origin over some [0, T], T < **,
within the class of L2(0, T; Z) controls v.
Then the solution operator P to the (ARE) guaranteed by Theorem 4.1
is an isomorphism on Y. •
Corollary 4.1
Under the assumptions of Theorem 4.2 the operator B 'P: Y --~ U is
bounded iff B: U ~ Y is bounded.
Corollary 4.2 [L-T.13 p. 43]
Assume the hypotheses of Theorem 4.1. In addition assume that the
free dynamics e At is a s.c group
uniformly bounded for negative times. Then the conclusion of
Corollary 4.1 applies.
As Corollary 4.2 shows, the property that the gain operator B°P is
unbounded is intrinsic to time
reversible dynamics under the hypothesis (I-I-2). We shall see
later that one "loses" this property for
classes of control problems (i.e. such that they do not satisfy
(H-2)) control operators which are treated
in the next section.
A different treatment of control problems with unbounded control
operators (subject to (H-2)) is via the
so called Dual Riccati Equations, as proposed by F. Flandoli. This
is to say that instead of looking at
the solution to (ARE) one considers the "Dual Riccati Operator" say
Q ~ .6 ~ which solves an
appropriate "Dual Riceati Equation." If the original dynamics is
represented by a group e At, then the
Dual Riccati Equation takes the form
(ARE-l) (AQ x,y)y + (QA*x,y)y - (B°x, B*y)u + (R*RQx, Qy)y = 0 x,y
~ ~9 (A*) c Y
The Dual Riccati Equation is simpler than (ARE) since the quadratic
term in (ARE-l) is now bounded.
The relation between ARE and ARE-1 in the case of group dynamics is
given by the following
Theorem.
Theorem 4.3 [F-L-T]
In addition to hypotheses of Theorem 4.1 we assume that A generates
a s.c. group and that both pairs
{A, B} and {A °, R*} are exactly controllable. Then there exists
unique solution Q E L (Y) to (ARE-l)
and
p'q = Q . •
Some extensions of the results of Theorem 4.3 to more general
dynamics are provided in [B.2] and [B- D].
Remark 4.1. Under the additional "smoothness" assumption imposed on
the observation R
T t IR, ReAtBulydt<CT [ul U
34
one obtains the additional regularity of the Riccati operator: B* P
e L (Y, U) (see [D-L-T], [L-T.13 p.
44]). However, the above assumption imposed on the observation R,
in the most interesting cases of
conservative hyperbolic dynamics, is in conflict with the
detectability condition and, ultimately, with the
stability of the feedback semigroup (see the results of Theorem 4.2
and Corollary 4.2). For this reason
the results leading to the bounded gain operator B*P (under
(H-2)-hypothesis) are of very limited
interest and will not be further pursued.
4.2. Fully unbounded control operators
In this section we shall dispense with regularity hypothesis (H-2).
Our motivation comes from several
hyperbolic and plate-like problems (see sec. S) where this
hypothesis is violated. Some results on the
wellposedness of Algebraic Riccad equations for the case of fully
unbounded control operators are
given by the following Theorem
Theorem 4.4 see [L-T.4]
In add/t/on to (1.7) and to the (F.C.C), assume that
(4.4) R*R is strictly positive
(4.5) The operator A* R*R A -1 ~ I"- CO-
Then there exists a positive selfadjoim solution P ~ £ (Y) of the
(ARE) such that
(4.6) B*P e f~CD(A); U)
(4.7) u°Ct; YO) = -]3* Py°(t; YO) e I..2(0, -- ; U)
(4.8) J (u °, yO) = (PYo, YO)y.
Moreover, P is the unique solution of (ARE) within the class of
selfadjoint operators P subject to
regularity requirements (4.6) •
In the present case the Riccati theory displays new features, or
pathologies, over the more regular case
when hypothesis (H-2) is in place. They include the following ones
(i) While in the case of assumption (H-2), the opt/real trajectory
y°(t) defines a s.c. semigroup no such
claim is now made. Indeed, in general y°(t) may be only a l-firne
integrated semigroup. No claim is made that the domain ~D (Ap) of
the generator Ap of 1-t/me integrated semigroup is dense in
Y.
(ii) While in the case of hypothesis (H-2), the s.c. semigroup e
Apt under detectability conditions (weaker then (4.4)) is
exponentially stable on Y, no such claim is now made in full
generality. Indeed, the original closed loop vrajectory y°(t) is
exponentially stable but for initial data Y0 contained in a
subspace which is strictly contained in Y see [I.,-T.4].
Off) In the case of hypothesis (H-2), the Riccati operator is an
isomorphism on Y when exact controllability (E.C.) holds (which in
our case (see (4.4)) holds for groups). Now, however P is bounded
and injective, but with possible unbounded inverse p--1 (sea
[L-T.4] for examples where not only is p--t unbounded but P changes
drastically the feedback dynamics from hyperbolic of e At
to parabolic, o f eAt=t).
3 5
As mentioned above, in general, the feedback semigroup need not
generate a s.c. semigroup on Y.
Ho~vever, for several specific dynamics one can show that this is
indeed the case i.e. the feedback
trajectory y0 is strongly continuous in Y. If this happens, uniform
decay as t --~ ** of the feedback
semigroup can be proved. This result is stated in the following
Theorem
Theorem 4.5 [L-T.4]
In addition to hypotheses of Theorem 4.4 we assume that for some T
> 0
(4.9) y°(t, 0; Y0) e C ([0,T]; Y).
Then the operator Ap = A - BB* P generates a strongly continuous
semigroup on Y and, moreover,
there exist constants C, co > 0 such that
l e ( A - B B ' P ) t l r ( y ) ~ C e " ~ t ; t > 0 . •
(4.10)
The cnax of the matter is, of course, to verify hypothesis (4.9).
In concrete example this is rather
technical p.d.e, question. In fact, as we shall see in the next
section, for several hyperbolic-like
problems, a positive answer is provided by applying P.D.E.
arguments based on rnicrolocal analysis and
pseudodifferential techniques.
5. Examples il lustrating the results of sub-section 4.1 (case of
H-2 hypothesis)
Here we shall provide several examples illustrating Theorems 4.1 -
4.3 which assume the validity of
hypothesis (H-2). In fact formulation of hypothesis (H-2) (see
[L-T.13]) was precisely motivated by the
discovery that it expresses sharp trace regularity of underlying
p.d.e, problems which hold true. The
abstract formulation of these trace regularity properties is
precisely (H-2).
5.1. Wave equation with boundary control [L-T.13, p. 71].
We consider the following problem
Wtt = AW ; in Q ;
(5.1) ~ w ( 0 , ' ) = w 0 ; w t ( 0 ; ' ) = w z ; i n f l ; /
Lwlz=u onE.
where we take the boundary control u e I.q(~). With (5.1) we
associate the cost functional (which is
motivated by regularity properties [L-T.9], [L-T. 10], [Lie.l],
[L-L-T]):
(5.2) J ( u , w ) = f [Iw(t)l 2 + I u(t) I ta2(r)] dt. b
la(n)
To put problem (5.1), (5.2) into abstract setting we introduce the
positive selfadjoim operator gh = -Ah;
~9 (.~) = HI(f2) ~ H2(~) and define the operators
36
A= I:D,I IA 01 and the space Z = Y = I.,2(f2) x H -I (f2) ; U =
(L2(F)). It is well known that A generates group on Y and
A-1Bu = [Du I is bounded: L2(I") --d, Y.
The crux of the matter is hypothesis (H-2). Indee..d, it can be
shown (see [L-T.2]) that (I-{-2) is
equivalent to the following inequality
(5.3) ~rf dE < CT [ l~bO [ 2 (t"2) + I ~b I 12L2(t.I)]
where ~b satisfies the homogene